Non-linear elasticity and some mathematical puzzles

The main source of non-linear elasticity and an-elastic behaviour is the mobility of microstruntures. The best studied class of materials in that of ferroelastics where the microstructure is related to pattern of twin boundaries including individual boundaries, needle shaped boundaries, S-shaped boundaries, tartan, and tweed structures. Their analysis in local and non-local approximations will be introduced.

Two novel developments have taken place: firstly we understand better that internal wall structures can exist. They can be decribed within the scheme of coupling theories involving elastic, electronic or chemical degrees of freedom. Secondly, local atomic-sized beeds of amorphous inclusions lead to macroscopic strain of the adjacent lattice and percolation behaviour for high concentrations. Their proper mathematical description is unknown but their physical behaviour becomes increasingly clear.